Lots of people know that the Riemann Hypothesis has something to do with prime numbers, but most introductions fail to say what or why. I’ll try to give one angle of explanation.

Layman’s Terms

Suppose you have a bunch of friends, each with an instrument that plays at a frequency equal to the imaginary part of a zero of the Riemann zeta function. If the Riemann Hypothesis holds, you can create a song that sounds exactly at the prime-powered beats, by simply telling all your friends to play at the same volume.

Mathematical Terms

Let $ \pi(x) $ denote the number of primes less than or equal to x. Recall Gauss’s approximation: $ \pi(x) \approx \int\_2\^x \frac{1}{\log t} \,dt $ (aka, the “probability that a number n is prime” is approximately $ \frac{1}{\log n} $).

we can again interpret it as an error-correcting formula for counting the primes.

Now since $ \psi(x) $ is a step function that jumps at the prime powers, its derivative $ \psi’(x) $ has spikes at the prime powers and is zero everywhere else. So consider

$$ \psi’(x) = 1 - \frac{1}{x(x\^2 - 1)} - \sum\_z x\^{z-1} $$

It’s well-known that the zeroes of the Riemann zeta function are symmetric about the real axis, so the (non-trivial) zeroes come in conjugate pairs $ z, \bar{z} $. But $ x\^{z-1} + x\^{\bar{z} - 1} $ is just a wave whose amplitude depends on the real part of z and whose frequency depends on the imaginary part (i.e., if $ z = a + bi $, then $ x\^{z-1} + x\^{\bar{z}-1} = 2x\^{a-1} cos (b \log x) $), which means $ \psi’(x) $ can be decomposed into a sum of zeta-zero waves. Note that because of the $2x\^{a-1}$ term in front, the amplitude of these waves depends only on the real part $a$ of the conjugate zeroes.

For example, here are plots of $ \psi’(x) $ using 10, 50, and 200 pairs of zeroes:

So when the Riemann Hypothesis says that all the non-trivial zeroes have real part 1/2, it’s hypothesizing that the non-trivial zeta-zero waves have equal amplitude, i.e., they make equal contributions to counting the primes.

In Fourier-poetic terms, when Flying Spaghetti Monster composed the music of the primes, he built the notes out of the zeroes of the Riemann zeta function. If the Riemann Hypothesis holds, he made all the non-trivial notes equally loud.

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